Probability and Tempered Modal Eliminativism

Abstract

In this paper the strategy for the eliminative reduction of the alethic modalities suggested by John Venn is outlined and it is shown to anticipate certain related contemporary empiricistic and nominalistic projects. Venn attempted to reduce the alethic modalities to probabilities, and thus suggested a promising solution to the nagging issue of the inclusion of modal statements in empiricistic philosophical systems. However, despite the promise that this suggestion held for laying the ‘ghost of modality’ to rest, this general approach, tempered modal eliminativism, is shown to be inadequate for that task.

Notes

Quine's well-known views are a good example of this kind of skepticism with respect to the modalities. See Quine ( 1960 , ch. 3, 1947 , 1953 ) for details.

See Beirmann and Faak 1957 , Hacking 1971 and Hacking ( 1975 , chs. 13–14).

This view constitutes a reductive elimination of probability, and depends on there being an isomorphism between the structure of probabilities (the algebra of probabilities) and the structure of possibilities (the algebra of possibilities), but the mapping from the elements of the algebra of probabilities to the elements of the algebra of possibilities is not a one-to-one mapping. As a result, the algebras are not isomorphic, and, hence, probabilities cannot be reduced to possibilities without loss of meaning.

The sense in which this program is an eliminative program, is in the sense of elimination via reduction. Probability claims are, in the Leibnizian view, completely eliminable slava veritate and without loss of meaning by systematically replacing them with the more fundamental modal claims in terms of which they are defined.

Now, Bigelow is clear that we cannot tell what the probability of a proposition is from the diameter of a proposition alone. Rather, he identifies them with a different technical concept associated with possible worlds; viz. volumes. See Bigelow ( 1976 , pp. 305–8) for more details. See Nilsson 1986 for a variant of this strategy.

This might straightforwardly seem to be the case as standard alethic modal operators are non-epistemic unary operators whereas probabilities are epistemic operators that are often conditional and dyadic, and this might indicate that probability is really an epistemic concept that depends on one's evidence and so is not a modality in the normal sense associated with the alethic modalities.

For historical details see Niiniluoto 1988a , Niiniluoto 1988b and Venn 1866 .

See, especially, Venn ( 1866 , ch. XII, § 20). Venn discusses several early attempts to rank modal concepts from the weakest to strongest, and he argues that such attempts were primitive or coarse modal attempts to do the work of the concept of probability in discrete terms. But, Venn superficially refers to the modalities in terms of degrees of certainty, and, as a result, one might be tempted to regard his project as only aimed at eliminating the epistemic modalities. However, it is clear that elimination of the modalities is intended to extend to the logical modalities and not just to epistemic modalities. Venn is clearly ambiguous about the nature of the modalities themselves as well as about the nature of logic, but in general his view of logic is very similar to Boole's view of logic as the laws of though (see Boole 1854 , Hailperin 1986 , 1996 ). Now, again, this might tempt one to interpret Venn's view of the modalities as epistemic as Niiniluoto ( 1988b ) does, but this is a rather uncharitable interpretation of Venn's work on logic. His work, by and large, predates the Fregean criticisms of psychologistic interpretations of logic, and so we can reasonably construe his view as the view that even the logical modalities are reducible to the probability calculus. This claim is least modestly supported in virtue of the fact that he is fully aware of the concept of mathematical certainty (see Venn 1866 , p. 317) which appears to be an important precursor element of the logicistic interpretation of logic.

For formal reasons one might immediately regard this intuition regarding the structural similarity between possibilities and probabilities as flawed. If reduction requires isomorphism, then there must be a one-to-one mapping from the algebra of possibilities to the algebra of probabilities. But, isomorphisms are symmetric only if there is a one-to-one mapping between them (i.e. they are structural equivalencies and require that the mapping F: A → B matches unique elements in A, x ∈ A, with unique elements in B, y ∈ B, and that the inverse map F ^− 1 maps unique elements of B into unique elements of A). As noted in note 3, there is no such mapping from the algebra of probabilities to the algebra of possibilities. Consequently, there can be no isomorphism between possibilities and probabilities. So, the Venn project can only be achieved if reduction does not require strict isomorphism, but rather some weaker relation like homomorphism, embedding or mere similarity. However, homomorphism, embedding and mere similarity are relations that are too weak to allow for eliminative reduction via definition. Homomorphisms are not symmetric, and, hence, although it is true that the algebra of possibilities is not homomorphic to the algebra of probabilities, it is true that the algebra of probabilities is homomorphic to the algebra of possibilities. This is interesting, because even if we relaxed the requirement for reduction from isomorphism to homomorphism, it would imply that modal eliminativism is false while leaving open the possibility of probabilistic eliminativism. But, this is no help to the nominalists and empiricists who want to eliminate the traditional modalities while retaining the concept of probability.

It also remains to be seen whether or not the criticisms of the Venn project presented herein can be extended to weaker modal concepts. This question will be left open for future investigations, but if the criticisms can be extended to weaker modalities, then there might be interesting consequences with respect to views like those espoused by D. H. Mellor ( 1995 , 2000). He claims, ‘The kind of necessity that “ch _c(E) = 1” must express is usually if not always contingent: it can exist in our world without existing in all other possible worlds and so cannot be metaphysical necessity [1.6] … But even if the necessity that “ch _c(E) = 1” expresses is weaker than metaphysical necessity, it must still entail existence’ (Mellor 1995 , p. 31). So, although Mellor is rather unclear what kind of modalities are employed in formulating the necessity condition he uses to explain causation in Mellor 1995 , if the modality in question is non-reducible to probability, then his view is in need of revision.

This must be the case because if ¬(a → b), then it is possible that (a & ¬b), and, in the case at hand, this would entail that there can be logical possibilities which are not non-zero probabilities. But, the reductionistic cum eliminativist Venn project requires that all modalities, including possibilities, must be reducible to probabilities. Modalities and probabilities must be completely co-referential if they are identical, and so if there is some proposition that is possible but has a zero probability, then modalities and probabilities are not co-referential. See Levi 1980 for discussion of the incompatibility of such claims, and Zaman 1987 for an attempt to deal with the compatibility of such claims.

It is important to notice that the probability referred to in this expression is an unconditional probability, and one might be tempted to claim that all probabilities are conditional probabilities. Often the conditional aspect of probabilities is omitted in probability expressions for convenience, and so we must be clear about the conditional or unconditional nature of probabilities. Now, we do not want to bias the issue either way, and we will attempt to be clear when considering the various interpretations of the probability calculus. In brief, empirical versions of objectivist interpretations hold that probabilities are not conditional on any source of information (i.e. are not conditional in the usual sense), while both other types of interpretations generally regard probability statements as conditional. In any case, one might be tempted to claim that it is obvious that ⋄p is compatible with P(p|e) = 0, but that ⋄(p|e) is not obviously compatible with P(p|e) = 0. This suggestion brings to mind von Wright's suggestion noted above concerning treating all modalities as dyadic, and, more controversially, as treating the alethic modalities as conditional on tautologous evidence (see von Wright 1989 , pp. 31–32). But, Venn does not appear to be aware of the concept of a conditional possibility, and is most concerned with the alethic modalities as opposed to, say, the epistemic modalities (which might, more plausibly, be regarded as conditional modalities). The view that the logical modalities can be seriously interpreted in conditional terms is unconvincing, and we will return to this topic in sections 2.1 and 2.2. But, Venn's project appears to be intended to eliminate logical possibility, etc., and not just epistemic possibility.

See van Fraassen 1989 , Lemmon 1977 , Konyndyk 1986 and Ackermann 1967 for discussions of permissible modal inferences and the problems associated with intensional logics. As Ackermann (1967, p. 17) points out, ‘it is similarly clear that ⋄p → p cannot also be a theorem of a modal calculus without making p and ⋄p provably equivalent, thus making the modal operator ⋄ redundant.’

As we shall see later, there is some question whether the logical theory of probability ought to be regarded as a species of the subjectivist theories or as a species of the objectivist theories. Ultimately it is probably best regarded as a species of OI.

For a survey of various ways of classifying interpretations of the probability calculus, and discussion of their adequacy see Weatherford ( 1982 , ch. 1). In terms of Weatherford's four fold distinction of theories of probability, OI would include the relative frequency (RF) and a priori (AP) theories, and SI would include the classical (CTP) and subjective (SUB) theories.

For the sake of simplicity assume that P(p) is definable, either as an absolute probability or as a conditional probability, but in general any reference to evidence and background knowledge will be omitted throughout.

I include discussion of subjective interpretations for the sake of both comprehensiveness and generality. In point of fact, Venn himself rejected subjective interpretations of the probability calculus, like that offered by De Morgan, as is clear from his comments (1962, pp. ix, 119n, 122–3). Nevertheless, as my aim to show the inadequacy of all forms of tempered modal eliminativism and not just the inadequacy of the Venn project, it is necessary to address such interpretations.

See Shimony ( 1955 , pp. 129–30).

See De Finetti 1972 , Ramsey 1926 and Levi 1978 .

For some more or less compelling examples of such views see Kahneman et al. 1982 , Cherniak 1986 , Kitcher 1992 , Smokler 1990 and Stich 1990 . And Quine's classic criticism of the analytic/synthetic distinction is found in Quine 1961 . For some objections to this view of the status of principles of rationality see Biro and Ludwig 1994 and Evnine 2001 .

The latter consideration is necessary to cover Popper's propensity interpretation of probabilities. On Popper's theory of probability probabilities are dispositions of physical objects.

Interestingly, as noted in the introduction, Venn has been credited (along with Peirce) as the founder of the relative frequency interpretation of probabilities, and this view is proposed in Venn 1866 . Consequently, one might plausibly interpret Venn's modal eliminativism more specifically as the claim that all the modalities can be eliminatively reduced to physical probabilities. However, this more controversial assumption is not presupposed here.

Hacking notes this ambiguity in Hacking ( 1975 , p. 149), and indicates that it is often presented as a criticism of logical theories of probability. Also, Jeffery 1975 acknowledges the ambiguous sense in which Carnap's theory of confirmation is meant to be logical.

See Carnap ( 1962 , pp. xv–xix) for an explanation of the ambiguity associated with the (a) explication.

This kind of theory is explicitly defended by Donald Gillies ( 1991 ), and Gillies argues that inter-subjective interpretations of probabilities are closely related to subjectivist interpretations of probabilities. On his view probability distributions can be attributed to social groups in states of consensus that are informationally transparent rather than to individuals, as in standard subjective interpretations. But, this view turns out to depend on contingent features of such social groups, and is equally susceptible to the criticisms directed at purely subjectivist interpretations of the probability calculus.

The point here is reminiscent of the point made concerning Kant's forms of sensible intuition as presented in The Critique of Pure Reason. The forms of sensible intuition, space and time, are regarded as necessary for experience, but the necessity involved here is relative and not absolute. The forms of sensible intuition may not hold objectively for all kinds of cognitive beings.

See Hilpinen 1975 .

Although this appears to be the intended interpretation of logical probabilities, it is not at all clear that such a concept makes sense. The empirical nature of what constitutes rational human behavior appears to undermine the possibility of grounding such putatively logical concepts.

Of course this presumes that we are talking about empirical claims, and that there really is one true description of reality. It may turn out, if the instrumentalists are correct and induction is unwarranted, that a full specification of the empirical facts would be compatible with several theories. This complication will be ignored here, however.

See Shimony ( 1955 , p. 127).

Curiously, the Leibnizian program of probabilistic eliminativism is also fundamentally misguided (although, formally, it is in better shape), but that is a topic for another paper. Presuming that this is the case, however, rejection of both the Venn project (modal eliminativism) and the Leibnizian project (probabilistic eliminativism) would show that the concepts of probability and of the traditional modalities are simply distinct.